相对整体维数有限的扩张
Extensions with Finite Relative Global Dimension
代数的扩张是指两个代数之间保持单位元的同态映射.设f:B→A是代数的扩张,扩张f的相对整体维数是指所有A-模的相对投射维数的上确界.我们给出了扩张的相对整体维数有限的一个充分必要条件,作为应用,还获得了Hochschild的文[Relative homological algebra,Trans.Am.Math.Soc.,1956,82:246–269]中一个结果的简洁证明.
An extension of algebras is a homomorphism of algebras preserving identities. The relative global dimension of an extension f:B→A is defined to be the supremum of relative projective dimensions of all A-modules. We give a necessary and sufficient condition for an extension of algebras to have finite relative global dimension. As an application, we give a new short proof for a result in Hochschild [Relative homological algebra,Trans. Am. Math. Soc., 1956, 82: 246–269].
相对投射模 / 相对投射维数 / 相对整体维数 {{custom_keyword}} /
relatively projective module / relative projective dimension / relative global dimension {{custom_keyword}} /
[1] Auslander M., Reiten I., Smalø S., Representation Theory of Artin Algebras, Cambridge University Press, Cambridge, 1995.
[2] Auslander M., Solberg Ø., Relative homology and representation theory Ⅱ, Relative cotilting theory, Comm. Algebra, 1993, 21: 3033–3079.
[3] Farnsteiner R., On the cohomology of ring extensions, Adv. Math., 1991, 87: 42–70.
[4] Green E. L., A criterion for relative global dimension zero with applications to graded rings, J. Algebra, 1975, 34: 130–135.
[5] Guo S. F., Relative global dimensions of extensions, Comm. Algebra, 2018, 46: 2089–2108.
[6] Hirata K., On relative homological algebra of Frobenius extensions, Nagoya Math. J., 1959, 15: 17–28.
[7] Hirata K., Sugano K., On semisimple extensions and separable extensions over non-commutative rings, J. Math. Soc. Japan, 1966, 18: 360–373.
[8] Hochschild G., Relative homological algebra, Trans. Am. Math. Soc., 1956, 82: 246–269.
[9] Nuss P., Noncommutative descent and non-abelian cohomology, K-Theory, 1997, 12: 23–74.
[10] Reiten I., Riedtmann Ch., Skew group algebras in the representation theory of artin algeras, J. Algebra, 1985, 92: 224–282.
[11] Thévenaz J., Relative projective covers and almost split sequences, Comm. Algebra, 1985, 13: 1535–1554.
[12] Xi C. C., On the finitistic dimension conjecture, I: related to representation-finite algebras, J. Pure Appl. Algebra, 2004, 193: 287–305; Erratum to “On the finitistic dimension conjecture, I: related to representationfinite algebras, J. Pure Appl. Algebra, 2004, 193: 287–305”, J. Pure Appl. Algebra, 2005, 202: 325–328.
[13] Xi C. C., On the finitistic dimension conjecture, Ⅱ: related to finite global dimension, Adv. Math., 2006, 201: 116–142.
[14] Xi C. C., Xu D. M., The finitistic dimension conjecture and relatively projective modules, Comm. Contemp. Math., 2013, 15: 1–27.
国家自然科学基金资助项目(11331006);广西自然科学基金项目(2018GXNSFAA138191);桂林航天工业学院博士基金项目(20180601-20200601)
/
〈 | 〉 |